Firstly, please accept my condolences on the passing of your wife's family member. May God console your wife and the whole family among the mourners of Zion and Jerusalem.
There is no actual obligation to say anything in particular aside from the traditional blessing of comfort (“Hamakom yenachem etchem b’toch shar avlei Tzion v’Yerushalayim” — May God comfort you amid all other mourners of Zion and Jerusalem). However, there is a custom that after these traditional words are said, someone then tells the mourners "Arise" - "Kumu”. The reason for this is that the mourner should show reluctance to get up from mourning, but should nevertheless be instructed to arise and continue living a purposeful life.
There is also a custom among some people to recite two verses: Isaiah 60:20 and 66:13.
“No more will your sun set, nor your moon be darkened, for God will be an eternal light for you, and your days of mourning will end.”
“Like a man whose mother consoles him, so too shall I (God) console you, and you shall be consoled in Jerusalem.”
There is also a custom that at the end of the seven-day shiva mourning period the mourners take a walk around the block. This is considered to be a symbolic act that represents the mourners entering back into society after a week of being at home during the shivah.
May you and your family merit to know only happy occasions from now on.
